Enhancing Conceptual Structures in the Parabola with 4T Models and Digital Technologies

1. 4T Models of Anthropological Moments of Didactical Praxeologies in the Parabola with Digital Technologies 1Clement Ayarebilla Ali 2Prof Ernest Kofi Davis, PhD 3Douglas Darko Agyei, PhD (1)Department of Basic Education, Faculty of Educational Studies, University of Education, Winneba, Ghana (2)(3)Department of Mathematics and Computer Science Education, Faculty of Science and Technology Education, University of Cape Coast, Cape Coast, Ghana

2. Outline of presentation • Introduction • 4T models of anthropological didactics • Methodology • Results • Conclusion and Recommendations • References • Appreciation

3. Introduction • Didactics [1, 2] is the development of conceptual structures based on the roles of human actions, reasoning, and discourses (praxeology). • Didactical praxeology [3, 4] is the division of didactics into praxis or know-how (actions, reasoning, activities, practices and discourses) and logos or know-why (discursive processes) in the two main blocks-- practical (i.e. tasks and techniques) and theoretical (theories and technologies). • Research [5, 6, 7] has extended the 3T models (tasks, techniques and theories) to the 4T models (tasks, techniques, technologies and theories). • The 4T models perform multiple roles, including but not limited to experimenting the two blocks, intertwining two ergonomics (i.e. instrumentation and instrumentalization in TPACK) with anthropological praxeologies, and establishing mathematics laboratories with anthropological moments of didactics [8, 9, 10]. • Even though these models exist for a very long time, teacher training and education and classroom discourses in Ghana and most developing countries sparingly explore these models. The 4T models therefore, explored the GeoGebra to address two research questions : 1. To what extent do the 4T models of anthropological moments of didactical praxeologies improve upon conceptual structures in the Parabola? 2. What didactical structures in the Parabola are most successful with the GeoGebra?

4. Figure 1: 4T Model of Anthropological Didactical Praxeologies

5. Experimental Methodology • The experimental methodology implored mixed method concurrent convergent design [11, 12] to explore both quantitative and qualitative data. • Quasi-experimental methods with digital technologies (GeoGebra) was used to collect quantitative data and unstructured interview guide was used to collect qualitative data after an initial pretest exercise was administered to thirty three groups of preservice teachers of the Department of Basic Education, University of Education, Winneba in Ghana [11, 12]. • Because the number of groups were less than 50, the Shapiro-Wilk test was used to test whether the quantitative data obtained was normally distributed, the paired samples t-test was performed to test whether there was a significant difference between the results of the pretest and post test exercises with SPSS 20 programme for Windows at 0.05 level of significance (two-tailed) [12]. • The qualitative data was analyzed by content analysis on the success or otherwise of without or with GeoGebra on the conceptual structures of the Parabola (focus, directrix, symmetry and vertex) [12]. • The mixed method design simultaneously triangulated the two methods, probed for further contradictory findings and provided multiple arguments of pragmatic views to ensure validity and reliability [11, 12].

6. Experimental Methodology cont’d Figure 2: A Model of Mixed Methods Concurrent Convergent Design [12]

7. Results and Discussion Research Question 1: To what extent do the 4T models of anthropological moments of didactical praxeologies improve upon conceptual structures in the Parabola? • The preservice teachers’ scores were statistically significant difference at Shapiro-Wilk test (pretest > 0.05; post test > 0.05), normally distributed and satisfied the condition for use of the paired samples t-test test of significant differences. Table 1: Paired Samples T-Test Results

8. Results cont’d • The results of paired samples t-test show that there is a statistically significant difference (t(32)=6.237, p=0.0001<0.05)) between pretest and post test exercises in terms of how the 4T models with GeoGebra improved preservice teachers’ conceptual structures. • Thus, while the conceptual structures test average of preservice teachers was 89.97 before the intervention with GeoGebra, this average rose sharply to 101.42 after the GeoGebra to confirm that adding digital technologies to the 3T models in anthropological moments of didactics has positive statistically significant effects in preservice teachers’ conceptual structures in the Parabola.

9. Results cont’d • Research Question 2: What didactical structures in the Parabola are successful with the GeoGebra most? • The study grouped conceptual structures with the dominant anthropological objects, and analyzed successes without and with GeoGebra in each conceptual structure. Table 2: Conceptual Structures and Dominant Objects

10. Results cont’d • Summary of the preservice teachers’ comments: • We didn’t know that focus, directrix symmetry and vertex can even be located on graphs with the GeoGebra software. 2. It is even far easy to compute for the focus, directrix symmetry and vertex as compared to the other methods 3. We find it easier and effective to use the software to find the focus, directrix, symmetry and vertex than using the formulas. 4. We have even discovered a lot of new things in the GeoGebra. 5. We will prefer to use the GeoGebra on all other mathematics topics.

11. Conclusions • The paired samples t-test show that there was a statistically significant difference (t(32)=6.237, p=0.0001<0.05))between pretest and post test exercises with the 4T models in the teaching and learning of the conceptual structures in the Parabola with GeoGebra. • The results of the content analysis also showed that even though preservice teachers seemed fairly successful in task and theory, and unsuccessful in technique and theory without the GeoGebra, they were at least very successful in all four conceptual structures. • Concurrently, we concluded that the deployment of the GeoGebra yielded both the statistically significant difference t-test results, and the successes in all the four conceptual structures.

12. Recommendations • The 4T models should be certified as one of the main instructional models for teaching and learning of mathematics at all levels of education. • Conceptual structures that are embedded in multiple mathematics domains should be explored with the 4T models. • Consensus efforts should be made to breakdown broad-based mathematics domains into smaller conceptual structures and explored with the 4T models.

13. References [1]. A. J. Bishop, H. Bauersfeld, J. Kilpatrick, C. Laborde, G. Leder, & S. Turnau. Theory of Didactical Situations in Mathematics, 1970–1990. Kluwer Academic Publishers: New York, (2002). [2]. Y. Chevallard, M. Bosch, S. Kim, (2015). What is a Theory According to the Anthropological Theory of the Didactic? In Marianna Bosch, Yves Chevallard, Ivy Kidron, John Monaghan and Hanna Palmér John Monaghan (eds.). CERME 9: Thematic Working Group 17, Theoretical perspectives and approaches in mathematics education research, (2015)p.35-45. [3].Y. Chevallard, Analyses praxéologiques: esquisse d’un exemple. IUFM, Toulouse, (2013). [4]. A. Clark-Wilson, G. Aldon, A. Cusi, M. Goos, M. Haspekian, O. Robutti., M. Thomas, The Challenges of Teaching Mathematics with Digital Technologies--The Evolving Role of the Teacher. In P. Liljedahl, C. Nicol, S. Oesterle, D. Allan, (Eds.). Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education, PME, Vancouver, (2014)Vol. 1p.177-206. [5]. C. Winsløw, Anthropological theory of didactic phenomena: Some examples and principles of its use in the study of mathematics education. In M. Bosch, J. Gascón, A. Ruiz Olarría, M. Artaud, A. Bronner, Y. Chevallard, G. Cirade, C. Ladage & M. Larguier (Eds.), Un panorama de TAD. CRM Documents 10. Centre de Recerca Matemàtica, Bellaterra, Barcelona, (2011)p. 117-138. [6]. K. Østergaard, Theory and practice in mathematics teacher education. IVe Congrès International sur la TAD, Toulouse, 21-26 April, 2013. [7]. M. Van den Heuvel-Panhuizen, P. Drijvers, M. Doorman, M. Van Zanten. Reflections from abroad on the Netherlands didactic tradition in mathematics education. Freudenthal Institute, Utrecht University, Utrecht, (2016). [8]. N.C. Presmeg, L. Radford, G. Kadunz, L. Puig, M. Roth, S. Prediger. Semiotics in mathematics education, 13th International Congress on Mathematical Education (ICME), Hamburg, 24-31 July, 2016. [9]. N. Sinclair, M. G. Bartolini Bussi, K. Jones, U. Kortenkamp, A. Leung, K. Owens, A. Zaslavsky, B. Ubuz, Geometry (including technology). 13th International Congress on Mathematical Education (ICME), Hamburg from 24-31 July, 2016. [10]. J. Trgalova, A. Clark-Wilson, H. G. Weigand, ‘Chapter 11: Technology and resources in mathematics education, ERME book, (2016). [11]. L. Cohen, L. Manion, K. Morrison, Research Methods in Education (9ed). Routledge/Taylor & Francis Group, New York, (2011). [12]. J. W. Creswell, Research design: Qualitative, quantitative, and mixed methods approaches (4th ed.). Thousand Oaks, CA: Sage Publications, (2014).

14. Appreciations • The Chairman/Chairperson • Keynote Speakers • Session Chairs • Prof. Ramón Sanguino Galván and Organizing Committee • Prof. Monica Martins and ICETIC Scientific Committee • Presenters and Participants of ICETIC Conference • Management and Staff of University of Education, Winneba • Management and Staff of University Cape Coast • Co-authors • Preservice Teachers as Research Participants

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